Quantitative Diagnosis of Water Resources Carrying Capacity Obstacle Factors Based on Connection Number and TOPSIS in Huaibei Plain

Abstract

To further quantitatively assess the water resources carrying capacity (WRCC) system and analyze and identify the regional water resources carrying state and the physical mechanism of the state change, WRCC and obstacle factor diagnosis were carried out. In this paper, we proposed the mobility matrix to determine the connection number components, considered the dynamic attributes of the difference degree coefficient, and calculated it using the semi-partial subtraction set pair potential and triangular fuzzy number, so as to construct the quantitative diagnosis method of regional WRCC obstacle factors based on the connection number and TOPSIS. The results applied to six cities in the Huaibei Plain showed that the WRCC fluctuated around grade 2 and was in a poor state, which was mainly due to the insufficient support force; the water resources carrying state of the six cities gradually improved from 2011 to 2018, but the state became worse in 2019, which was related to the low precipitation in that year, the reduction in water resources, and the high degree of water resource utilization. The WRCC of Fuyang and Huainan was worse than that of the other four cities; over the 9 years, the average grades of Fuyang and Huainan were 2.26 and 2.43, while those of Huainan, Bozhou, Suzhou, and Bengbu were 2.19, 2.12, 2.05, and 2.05, respectively. The key obstacles limiting the improvement in the WRCC of the Huaibei Plain were per capita water resources, annual water production modulus, per capita water supply, vegetation coverage ratio, utilization ratio of water resources, water consumption per 10⁴ yuan value-added by industry, and population density. In time, the key obstacle factors in neighboring years generally tended to have similarity, and conversely appeared as a difference; in space, neighboring regions showed similarity and conversely presented as a difference. The results of this study can offer technical support and a decision-making basis for water resources management in the Huaibei Plain. The method constructed in this paper is extremely interpretive, easy to calculate, highly sensitive, and reliable in application results, which opens up a new perspective for the rational determination of the connection number and the difference degree coefficient and provides a new intelligent way to determine the state of a complex set pair system and its causal mechanism analysis and diagnosis of obstacle factors.

1. Introduction

From the perspective of physical causes, the water resources carrying capacity (WRCC) is the dynamic balance between water resources carrying support and pressure force under regulation force [1,2]. The obstacle factors can be understood as elements or indexes that lead to a poor WRCC [3,4,5]. WRCC evaluation and obstacle factor diagnosis are a quantitative water resources security measurement and early warning tool [6], which is crucial to the optimal regulation of water resources and sustainable development [7,8], especially for water-scarce regions, such as the Huaibei Plain, China.

WRCC evaluation is an emerging multidisciplinary and interdisciplinary research field that has been developing rapidly in the past 30 years [9,10], and its difficulties include uncertainty, dynamics, and quantification [11,12,13]. Ren et al. [14] employed the analytic hierarchy process and the entropy method to evaluate the WRCC. Yang et al. [15] estimated the WRCC using an “inversion model”. Wang et al. [16] constructed a decision-support framework to analyze the risk of the WRCC. Cao et al. [17] established a method based on the cloud model theory aimed at WRCC evaluation. Jin et al. [18] conducted WRCC evaluations adopting the connection number and set pair potential.

The main methods of evaluation of the WRCC included the analytic hierarchy process and entropy method, the decision support framework, the cloud model theory, and the connection number (CN) method. Among the above-mentioned, the CN method, which is able to quantitatively analyze the relation between the index value and the evaluation grade, was extensively adopted and can comprehensively and completely quantitatively portray the attributes of a set pair system from similarity, difference, and opposition perspectives, well solving the evaluation problem of a single index and providing a proven approach for WRCC evaluation, which has received the attention of many researchers and has achieved a large number of results [19,20,21]. Cui et al. [22] proposed an evaluation model by applying connection numbers. Kang et al. [23] constructed a five-element connection number model to evaluate the WRCC. Nevertheless, only a few of these studies have focused on plains agricultural areas. In addition, the reasonable calculation of the connection number components is the key to the CN method. The existing methods for determining the connection number components only pay attention to the connection degree between the index value and the evaluation grade from a macroscopic view, while less consideration is given to the influence of microscopic movements among the components. Some studies have ignored this effect and have directly used components for problem-specific analyses, such as calculating the evaluation grade based on the attribute identification method or maximum affiliation [21,23,24], which obviously may produce large errors.

Using the connection number method for comprehensive evaluation of the WRCC, it was also crucial to convert the connection number containing information in certainty and uncertainty into a connection number value that is certain [25]. The challenge to determine this value was how to accurately calculate the difference degree coefficient I. At present, the methods in determining the difference degree coefficient I in the connection number mainly include the traditional valuation method [26], the gray correlation degree method [27], and the triangular fuzzy number method [28]. These methods for determining the I are inclined to be static without considering the dynamic characteristics of the I with the evaluated sample’s value. For example, the traditional valuation method for ternary connection number takes I = 0 [26], and the triangular fuzzy number method uses I = (−0.5, 0, 0.5) for the difference degree coefficient [28], which has a significant impact on the quantitative analysis of uncertain systems, such as water resources. In addition, there are few studies on I. Therefore, a method for I, consistent with the actual physical meaning, is highly needed.

Obstacle factor diagnosis is the follow-up of the WRCC evaluation and an important early warning and regulation tool. Sun et al. [6] analyzed the key factors of the WRCC using the obstacle degree model (ODM). Yuan et al. [4] revealed the obstacles to the ODM. Sun et al. [29] examined the WRCC of YRB adopting the ODM. Wang et al. [30] conducted an ODM to diagnose the obstacles to urban ecological civilization. Obstacle factor diagnosis was widely researched in the resources and environment, and it was the way to deeply understand and grasp the primary contradiction of these systems, which is helpful for the precise regulation of system elements to achieve a sustainable development of resources. However, the majority of research has adopted the ODM for diagnosis, which only considers the index value and has not yet taken into account the uncertainty relationship between the index value and the evaluation grade, and has insufficient physical significance, poor interpretability, and biased calculation process and results towards static. Hence, further enriching and improving the quantitative obstacle factor diagnosis method was an urgent problem to be solved, thus forming a set of evaluation–diagnosis–regulation technical models for WRCC system analysis. In addition, few research results have been reported on identifying the obstacle factors for the WRCC in Huaibei Plain.

This paper takes the WRCC obstacle factor as the research object and focuses on the three major problems that exist above: (1) how to quantitatively portray the migration and movement between the connection number components and reasonably determine it to achieve an accurate evaluation of the WRCC; (2) how to realize the dynamic value of the difference degree coefficient so that it changes with the change in the evaluation sample; (3) how to further enrich the theoretical and intelligent methods for identifying the obstacle factors to overcome the traditional obstacle degree model (ODM) with its shortcomings, such as lack of physical significance, poor interpretability, biased calculation process and results towards static, etc. In this work, we constructed 13 evaluation indexes and evaluation grade criteria of the WRCC, proposed a mobility matrix to reflect the microscopic movement among the connection number components, and thus build the method, and corrected the connection number component value, to evaluate the WRCC in Huaibei Plain. Meanwhile, a method based on the semi-partial subtraction set pair potential and triangular fuzzy number dynamic determination of the difference degree coefficient was constructed to obtain the single index set pair potential value, and an attempt was made to calculate the obstacle degree of the indexes by combining the CN and TOPSIS to parse out the physical mechanism of the change in the carrying state and to analyze the similarities and differences of the key obstacle factors in time and space. A quantitative diagnosis method of the WRCC obstacle factor based on the CN and TOPSIS was formed and applied to Huaibei Plain, China. The evaluation and diagnosis results can offer a foundation for water resources management in Huaibei Plain.

2. Materials and Methods

To quantitatively evaluate the WRCC and accurately diagnose the obstacle factors based on the CN and TOPSIS, a quantitative diagnostic method consisting of seven steps was constructed, as shown in Figure 1.

Step 1: From the perspective of the physical causes of the WRCC, combined with the existing literature, the previous research results of our laboratory, expert consultation, the characteristics of the study area, and the results of the survey, 13 evaluation indexes, including 3 subsystems (the support, regulation, and pressure force subsystems of the WRCC) were constructed [22,31], forming the index system {x_jk|j = 1, 2, …, n_k; k = 1, 2, 3} and evaluation grade criteria {s_gj|g = 1, 2, …, n_g; j = 1, 2, …, n_j}. The evaluation index value set was denoted as {x_ijk|i = 1, 2, …, n_i; j = 1, 2, …, n_k; k = 1, 2, 3. x_ijk is the value of i sample k subsystem j index of the WRCC, n_i, n_g are the evaluation sample number and grade number, respectively, and n_k is the index number in the k-th subsystem. This paper took 3 evaluation grades, i.e., g = 1, 2, and 3; grades 1, 2, and 3 represent “loadable”, “critically overloaded”, and “overloaded”, respectively [22].

Step 2: Uncertainty is a widespread phenomenon in nature and human society, and it has long been hoped that uncertainty can be converted into certainty under certain conditions, aiming to explore the underlying principles of uncertainty. The theory of set pair analysis (SPA), proposed by the Chinese scholar Zhao Keqin in 1989 [26], is a system theory that associates certainty with uncertainty, and can quantitatively describe and deal with certainty and uncertainty. The basic idea of SPA is to establish a set pair from two sets that are related in some way and to analyze the elements of the set pair in terms of similarity, difference, and opposition, so as to establish the CN to characterize the connection degree between the sets. Ternary CN can be written as follows [26]:

$$\mu =a+bI+cJ,$$

(1)

where a, b, and c represent the similarity, difference, and opposition degree components, respectively.

I, the difference degree coefficient, is able to transform the b (uncertainty) into the a or c (certainty) components under certain conditions, enabling uncertainty analysis. J is the opposition degree coefficient, and J = −1. The set pair is the basic structure for calculating the CN. In this study, x_ijk and s_gj were constructed as a set pair. For the positive (negative) index, grade g increases (decreases) with the increase (decrease) in x_ijk, and the ternary CN calculation formula is as follows [32,33]:

$$\left\{\begin{array}{l}{u}_{ijk1}=1\hfill \\ u_{ijk2}=\begin{array}{ll}1-2(s_{1j}-x_{ijk})/(s_{1j}-s_{0j}),& \begin{array}{l}{s}_{0j}<x_{ijk}\le s_{1j}\mathrm{for}\mathrm{positive}\mathrm{index},\\ \mathrm{or}{s}_{0j}>x_{ijk}\ge s_{1j}\mathrm{for}\mathrm{negative}\mathrm{index}\end{array}\end{array}\hfill \\ u_{ijk3}=-1\hfill \end{array}\right.$$

(2)

$$\left\{\begin{array}{l}{u}_{ijk1}=1-2(x_{ijk}-s_{1j})/(s_{2j}-s_{1j})\hfill \\ u_{ijk2}=\begin{array}{ll}1,& \begin{array}{l}{s}_{1j}<x_{ijk}\le s_{2j}\mathrm{for}\mathrm{positive}\mathrm{index},\\ \mathrm{or}{s}_{1j}>x_{ijk}\ge s_{2j}\mathrm{for}\mathrm{negative}\mathrm{index}\end{array}\end{array}\hfill \\ u_{ijk3}=1-2(s_{2j}-x_{ijk})/(s_{2j}-s_{1j})\hfill \end{array}\right.$$

(3)

$$\left\{\begin{array}{l}{u}_{ijk1}=-1\hfill \\ u_{ijk2}=\begin{array}{ll}1-2(x_{ijk}-s_{2j})/(s_{3j}-s_{2j}),& \begin{array}{c}\begin{array}{l}{s}_{2j}<x_{ijk}\le s_{3j}\mathrm{for}\mathrm{positive}\mathrm{index},\\ \mathrm{or}{s}_{2j}>x_{ijk}\ge s_{3j}\mathrm{for}\mathrm{negative}\mathrm{index}\end{array}\end{array}\end{array}\hfill \\ u_{ijk3}=1\hfill \end{array}\right.$$

(4)

where s_1j and s_2j are the critical values of evaluation grades 1 and 2, and grades 2 and 3, respectively. s_0j and s_3j denote the other critical values of evaluation grades 1 and 3, respectively. Equations (2)–(4) can also be represented by Figure 2 and Figure 3.

Taking the positive index as an example, when the positive index value x_ijk for the WRCC falls in (s_0j, s_1j), then the connection degree between the x_ijk and grade 1 is 1 (u_ijkg = 1), which means that the possibility of belonging to grade 1 is greater; the connection degree with grade 2 is a monotonic increasing function that increases with the increase in x_ijk; the connection degree with grade 3 is the smallest. The CN can quantitatively portray the relationship between the index value and the grade from three perspectives, which supplies an available technical approach for single index evaluation.

The relative difference degree function between x_ijk and s_gj is represented by u_ijkg, u_ijkg ∈ [−1, 1], and the corresponding relative affiliation degree is as follows [34]:

$$v_{ijkg}^{*}=0.5+0.5u_{ijkg}.$$

(5)

Thus, $v_{ijkg}^{*}$ in Equation (5) is normalized to the gain connection number component v_ijkg for the WRCC [35]:

$$v_{ijkg}=\frac{v_{ijkg}^{*}}{v_{ijk1}^{*}+v_{ijk2}^{*}+v_{ijk3}^{*}}.$$

(6)

u_ijk, the single index CN, can be formed from the connection number component v_ijk [36]:

$$u_{ijk}=v_{ijk1}+v_{ijk2}I+v_{ijk3}J.$$

(7)

Step 3: There are always different levels of mutual relationships between the connection number components. Nevertheless, less consideration was given to the influence of microscopic movements among the connection number components of index in existing studies, and these components are immediately employed for analysis of concrete issues, such as calculating the evaluation grade directly based on the attribute identification method or the maximum affiliation, which obviously may cause large errors. So, to quantitatively characterize the microscopic movements among the connection number components, explore the structure of the connection number system and its dynamic evolution mechanism, and more accurately determine the components of the connection number and the grade of the WRCC, from the perspective of the complex system structure of the set of WRCC evaluation samples and evaluation grade criteria, we applied a partial connection number to construct a mobility matrix that can quantitatively portray the microscopic movements among the connection number components and proposed a method to calculate it. This work has rarely been seen before. According to the adjoint function, especially the partial connection number, the CN and partial connection number are applied to describe the mutual movement between components and then obtain the corrected connection number components, which further enhances the reasonableness and accuracy of WRCC evaluation.

This paper defined the mobility matrix X for u = a + bI + cJ based on the partial connection number; X is as follows [20,37]:

$$X=\left[\begin{array}{ccc}1& {\partial }^{-}b& {\partial }^{-}b{\partial }^{-}c\\ {\partial }^{+}a& 1& {\partial }^{-}c\\ {\partial }^{+}b{\partial }^{+}a& {\partial }^{+}b& 1\end{array}\right]=\left[\begin{array}{ccc}1& \frac{b}{a+b}& \frac{b}{a+b}\frac{c}{b+c}\\ \frac{a}{a+b}& 1& \frac{c}{b+c}\\ \frac{b}{b+c}\frac{a}{a+b}& \frac{b}{b+c}& 1\end{array}\right].$$

(8)

The matrix X reveals the law of mutual transformation of the components. U = [a, b, c] is multiplied with matrix X, i.e., the connection number components are transformed by matrix X to obtain R. The connection number matrix R is as follows [20]:

$$\begin{array}{l}R=UX=[a,b,c]\left[\begin{array}{ccc}1& \frac{b}{a+b}& \frac{b}{a+b}\frac{c}{b+c}\\ \frac{a}{a+b}& 1& \frac{c}{b+c}\\ \frac{b}{b+c}\frac{a}{a+b}& \frac{b}{b+c}& 1\end{array}\right]\\ =\left[a+b\frac{a}{a+b}+c\frac{b}{b+c}\frac{a}{a+b}, b+a\frac{b}{a+b}+c\frac{b}{b+c}, c+b\frac{c}{b+c}+a\frac{b}{a+b}\frac{c}{b+c}\right]\end{array}.$$

(9)

Applying Equation (9) to v_ijk1, v_ijk2, and v_ijk3 in Equation (7), we obtained the following equation [20]:

$$\begin{array}{l}[v_{ijk1},v_{ijk2},v_{ijk3}]\left[\begin{array}{ccc}1& \frac{v_{ijk2}}{v_{ijk1}+v_{ijk2}}& \frac{v_{ijk2}}{v_{ijk1}+v_{ijk2}}\frac{v_{ijk3}}{v_{ijk2}+v_{ijk3}}\\ \frac{v_{ijk1}}{v_{ijk1}+v_{ijk2}}& 1& \frac{v_{ijk3}}{v_{ijk2}+v_{ijk3}}\\ \frac{v_{ijk2}}{v_{ijk2}+v_{ijk3}}\frac{v_{ijk1}}{v_{ijk1}+v_{ijk2}}& \frac{v_{ijk2}}{v_{ijk2}+v_{ijk3}}& 1\end{array}\right]\\ =[v_{ijk1}+v_{ijk2}\frac{v_{ijk1}}{v_{ijk1}+v_{ijk2}}+v_{ijk3}\frac{v_{ijk2}}{v_{ijk2}+v_{ijk3}}\frac{v_{ijk1}}{v_{ijk1}+v_{ijk2}},v_{ijk2}+v_{ijk1}\frac{v_{ijk2}}{v_{ijk1}+v_{ijk2}}+v_{ijk3}\frac{v_{ijk2}}{v_{ijk2}+v_{ijk3}},v_{ijk3}+v_{ijk2}\frac{v_{ijk3}}{v_{ijk2}+v_{ijk3}}+v_{ijk1}\frac{v_{ijk2}}{v_{ijk1}+v_{ijk2}}\frac{v_{ijk3}}{v_{ijk2}+v_{ijk3}}]\end{array}.$$

(10)

Then, each connection number component in Equation (10) after normalization to get the corrected single index CN $u_{ijk}^{\prime }$ [36,38]:

$$u_{ijk}^{\prime }=v_{ijk1}^{\prime }+v_{ijk2}^{\prime }I+v_{ijk3}^{\prime }J$$

(11)

where $v_{ijk1}^{\prime }$ is the similarity degree component, $v_{ijk2}^{\prime }$ is the difference degree component, and $v_{ijk3}^{\prime }$ is the opposition degree component transformed by the mobility matrix. According to Equation (11), the CN of sample i of the WRCC is computed as follows [36,38]:

$$u_i^{\prime }=v_{i1}^{\prime }+v_{i2}^{\prime }I+v_{i3}^{\prime }J={\displaystyle \sum _{j=1}^{n_k}{\displaystyle \sum _{k=1}^3{w}_j{v}_{ijk1}^{\prime }}}+{\displaystyle \sum _{j=1}^{n_k}{\displaystyle \sum _{k=1}^3{w}_j{v}_{ijk2}^{\prime }}}I+{\displaystyle \sum _{j=1}^{n_k}{\displaystyle \sum _{k=1}^3{w}_j{v}_{ijk1}^{\prime }}}J$$

(12)

where w_j is the weight of the j-th index.

Step 4: Based on the CN in step 3, the level eigenvalue method and the attribute identification method, which calculate the water resources carrying capacity, are as follows [34,39]:

$$H_i=1v_{i1}^{\prime }+2v_{i2}^{\prime }+3v_{i3}^{\prime }$$

(13)

$$h_i=\min \left\{\left.g\right|{\displaystyle \sum _{g=1}^g{v}_{ig}^{\prime }\ge \lambda }\right\}$$

(14)

where H_i and h_i are the evaluation samples i of the WRCC grade value; λ is the confidence degree, generally taking a value within [0.5, 0.7]. The larger λ is, the more reliable the evaluation results. We took λ = 0.6 in this paper.

Step 5: According to the set pair analysis theory and connection number expression (1), the difference degree coefficient I is a bridge between the difference degree b and the similarity degree a and the opposition degree c, which is an important interface between the connection number theoretical model and the actual research problem [25]. However, at present, the methods that include the traditional valuation method [26], the gray correlation degree method [27], and the triangular fuzzy number method [28] for determining the difference degree coefficient I are inclined to be static without considering the dynamic characteristics of the I changing with the evaluated sample’s value. For example, the traditional valuation method for the ternary connection number took I = 0 [26], and the triangular fuzzy numbers method had I = (−0.5, 0, 0.5) for the difference degree coefficient [28], which have a significant impact on the quantitative analysis of uncertain systems, such as the water resources system. In addition, there are few studies on the I. Therefore, there is a great need to establish a method for the I that is consistent with the actual physical meaning.

The I is the core of the quantitative description of uncertainty in a set pair at the microscopic level, and an important source of that uncertainty is the information contained in the sample value of the research problem. The physical significance of the I can be interpreted as the degree of transformation of the difference degree b into the similarity degree a or the opposite degree c, and the direction of the transformation (into a or c) and the degree of the transformation should be closely related to the proximity of the sample value to the criteria grade, i.e., the larger the value of a (or c), the more b is transformed into a (or c) [18,25]. Therefore, the I should be dynamic with the connection number components (a, b and c) of the sample, i.e., dynamic with the actual sample value of the problem under discussion.

Based on the semi-partial subtraction set pair potential ($s_f(u)=v_{ijk1}^{\prime }+v_{ijk2}^{\prime }(\frac{v_{ijk1}^{\prime }}{v_{ijk1}^{\prime }+v_{ijk2}^{\prime }}-\frac{v_{ijk3}^{\prime }}{v_{ijk2}^{\prime }+v_{ijk3}^{\prime }})-v_{ijk3}^{\prime }$), this study considered that the most possible value of the difference degree coefficient of the single index ternary connection number for the evaluation of the regional water resources carrying capacity is $\frac{v_{ijk1}^{\prime }}{v_{ijk1}^{\prime }+v_{ijk2}^{\prime }}-\frac{v_{ijk3}^{\prime }}{v_{ijk2}^{\prime }+v_{ijk3}^{\prime }}$, the minimum possible value is $-\frac{v_{ijk3}^{\prime }}{v_{ijk2}^{\prime }+v_{ijk3}^{\prime }}$, and the maximum possible value is $\frac{v_{ijk1}^{\prime }}{v_{ijk1}^{\prime }+v_{ijk2}^{\prime }}$, and the above three values can construct the triangular fuzzy number of the difference degree coefficient. Therefore, it is feasible to construct a semi-partial subtraction set pair potential (SSSPP) and triangular fuzzy number method to solve the I. Obviously, this method has more physical meanings and takes into account the dynamic change attributes of the I.

Comprehensive analysis of the above for the single index connection number of the evaluation of the regional water resources carrying capacity ($u_{ijk}^{\prime }=v_{ijk1}^{\prime }+v_{ijk2}^{\prime }I+v_{ijk3}^{\prime }J$) in Equation (11), we can obtain the triangular fuzzy number of the difference degree coefficient I, according to the $s_f(u)=v_{ijk1}^{\prime }+v_{ijk2}^{\prime }(\frac{v_{ijk1}^{\prime }}{v_{ijk1}^{\prime }+v_{ijk2}^{\prime }}-\frac{v_{ijk3}^{\prime }}{v_{ijk2}^{\prime }+v_{ijk3}^{\prime }})-v_{ijk3}^{\prime }$, as follows [18,25]:

$$\stackrel{\sim }{\xi }=({\xi }_1,{\xi }_2,{\xi }_3)=(-\frac{v_{ijk3}^{\prime }}{v_{ijk2}^{\prime }+v_{ijk3}^{\prime }},\frac{v_{ijk1}^{\prime }}{v_{ijk1}^{\prime }+v_{ijk2}^{\prime }}-\frac{v_{ijk3}^{\prime }}{v_{ijk2}^{\prime }+v_{ijk3}^{\prime }},\frac{v_{ijk1}^{\prime }}{v_{ijk1}^{\prime }+v_{ijk2}^{\prime }})$$

(15)

where s_f(u) is the semi-partial subtraction set pair potential (SSSPP); ξ₁, ξ₂, and ξ₃ are the minimum possible value, the most possible value, and the maximum possible value of the difference degree coefficient I, respectively.

If an intercept level α (0 ≤ $\alpha$ ≤ 1) is given, $\stackrel{\sim }{\xi }$ can be transformed into an interval corresponding to α as follows [28]:

$$\stackrel{\sim }{{\xi }_{\alpha }}=[\stackrel{\sim }{{\xi }_{\alpha }^L},\stackrel{\sim }{{\xi }_{\alpha }^R}]=[({\xi }_2-{\xi }_1)\alpha +{\xi }_1,-({\xi }_3-{\xi }_2)\alpha +{\xi }_3]=[\frac{v_{ijk1}^{\prime }}{v_{ijk1}^{\prime }+v_{ijk2}^{\prime }}\alpha -\frac{v_{ijk3}^{\prime }}{v_{ijk2}^{\prime }+v_{ijk3}^{\prime }},-\frac{v_{ijk3}^{\prime }}{v_{ijk2}^{\prime }+v_{ijk3}^{\prime }}\alpha +\frac{v_{ijk1}^{\prime }}{v_{ijk1}^{\prime }+v_{ijk2}^{\prime }}]$$

(16)

where α is the intercept level, usually taking α = 0.75. Obviously, Equation (16) realizes that the difference degree coefficient I changes dynamically with the evaluation sample’s value. The I in Equation (16) is brought into the single index connection number $u_{ijk}^{\prime }$ in Equation (11) to obtain the connection number value interval, and here the expected value of the connection number value interval is taken as a single index set pair potential value $s_{ij}\left(u\right)$.

The I is the key to compute the connection number value, and its essence is to construct a connection number adjoint function $f(v_{ijk1}^{\prime },v_{ijk2}^{\prime },v_{ijk3}^{\prime })$. Therefore, exploring the value of I is an important frontier in the research of the connection number adjoint function theory.

Step 6: Obstacle factors are elements or indexes that contribute to a poor WRCC. Obstacle factor diagnosis is the follow-up of WRCC evaluation and an important early warning and regulation tool [40]. The results of the diagnosis contribute to water resources management. However, the majority of research adopts the obstacle degree model (ODM) for obstacle factor diagnosis at present [4,6,29,30]. Aiming at overcoming the shortcomings of the obstacle degree model, such as only considering the value of the index, insufficient physical significance, poor interpretability, and biased calculation process and results towards static, this paper investigated an intelligent method for the quantitative diagnosis of obstacle factors based on the connection number and TOPSIS.

The TOPSIS method, also known as the ideal solution [41], is based on the relative proximity between the evaluation object and the ideal solution, ranking the target evaluation object to determine the superior and inferior evaluation objects. The closer the evaluated object is to the positive ideal solution, and the further it is from the negative ideal solution, the better the evaluation result of the evaluated object. If the object is closer to the negative ideal solution and further away from the positive ideal solution, the poorer the evaluation result of the evaluated object.

Based on the above principles, an attempt can be made to combine the connection number and TOPSIS to diagnose the obstacle factor of the WRCC. In this paper, the ideal solution was skillfully determined by calculating the subtraction set pair potential value ($s_{ij}\left(u\right)$) of the evaluation index of the WRCC and analyzing its range, from which the distance between each evaluation index and the ideal solution was calculated, and the obstacle degree of each index was finally obtained. This work has rarely been reported before.

The first step is to determine the best level and the worst level (positive and negative ideal solutions, respectively) of the set pair potential value of the evaluation index of the WRCC, and then calculate the distance between the set pair potential value and the positive and negative ideal solutions on the basis of considering the weight of the index, reflecting the superior and inferior of the index via distance. Finally, the obstacle degree of each evaluation index of the WRCC is obtained. When the distance between the set pair potential value and the positive ideal solution is larger, then the index is inferior, i.e., the greater the obstacle degree; when the distance between the set pair potential value and the positive ideal solution is smaller, then the index is superior, i.e., the smaller the obstacle degree.

The TOPSIS method can quantify the distance between each evaluation index and ideal solution, reflecting the superior and inferior of the WRCC evaluation indexes, easily calculating and helping to diagnose the obstacle factors accurately and dynamically.

In the TOPSIS method, the subtraction set pair potential value ($s_{ij}\left(u\right)$) of the evaluation index of the WRCC should be normalized, and $s_{ij}\left(u\right)$ is linearly transformed as follows:

$$s_{ij}^{\prime }(u)=0.5s_{ij}(u)+0.5.$$

(17)

The distance between the set pair potential value of sample i index j and the ideal solutions can be determined using Equations (18) and (19) [42,43]:

$$d_{ij}^{+}=w_j\sqrt{{(s_{ij}^{\prime +}(u)-s_{ij}^{\prime }(u))}^2}$$

(18)

$$d_{ij}^{-}=w_j\sqrt{{(s_{ij}^{\prime -}(u)-s_{ij}^{\prime }(u))}^2}$$

(19)

where $s_{ij}^{\prime +}(u)$ and $s_{ij}^{\prime -}(u)$ are the positive and negative ideal solutions of the evaluation index of the WRCC, respectively, $s_{ij}(u)$ is the subtraction set pair potential value of the evaluation index of the WRCC, and $s_{ij}^{\prime }(u)$ is the value obtained via linear transformation of the $s_{ij}(u)$. Since $s_{ij}^{\prime }(u)$ ∈ [0, 1], when $s_{ij}^{\prime }(u)$ = 0, it means that the evaluation index of the WRCC is in the worst state, and when $s_{ij}^{\prime }(u)$ = 1, it means that the evaluation index of the WRCC is in the best state. Therefore, the negative ideal solution is $s_{ij}^{\prime }(u)$ = 0, which is denoted as $s_{ij}^{\prime -}(u)$, and the positive ideal solution is $s_{ij}^{\prime }(u)$ = 1, which is denoted as $s_{ij}^{\prime +}(u)$.

Step 7: The obstacle degree of index j of WRCC sample i can be expressed as follows [44]:

$$O{b}_{ij}=\frac{d_{ij}^{+}}{d_{ij}^{+}+d_{ij}^{-}}.$$

(20)

where $d_{ij}^{+}$($d_{ij}^{-}$) represents the distance between the evaluation index and the positive (negative) ideal solution. Obviously, the more Ob_ij tends to 1, the greater the distance of the evaluation index deviating from the positive ideal solution, i.e., the index is the WRCC obstacle factor, which is in urgent need of human intervention and regulation. This method not only provides a new intelligent way for the diagnosis of obstacle factors, but also dynamically identifies the distribution characteristics and spatial–temporal scale change rules of the obstacle factors.

3. Study Area

The Huaibei Plain in Anhui Province, one of the three major plains in China, is located in eastern China and belongs to the North China Plain. This region includes six cities, namely Bengbu, Huainan, Fuyang, Huaibei, Bozhou, and Suzhou, as shown in Figure 4. In 2019, the administrative area of the entire Huaibei Plain was 10,348 km², accounting for 35.63% of the province’s (29,266 km²); the total resident population is 28,394,000, which accounts for 44.60% of the province’s (63,659,000); the arable land area is about 2.70 × 10⁶ hm², making up 48.65% of the province’s (5.55 × 10⁶ hm²). The multi-year average total water resources are 125.60 × 10⁸ m³, accounting for only 17.02% of the province’s (738.15 × 10⁸ m³), and the per capita water resources of each city account for less than 40% of the province’s, with the typical characteristics of a large population, a large area, and small water resources. A reasonable evaluation of the WRCC for each city in Huaibei Plain and a diagnosis of the WRCC obstacle factors are urgently required.

4. Results and Discussion

4.1. Evaluation of the Water Resources Carrying Capacity

The evaluation index system and evaluation grade criteria of the WRCC were established according to step 1, and the weights were gained using the fuzzy analytic hierarchy process based on the accelerated genetic algorithm (AGA-FAHP) [18,35,45], as shown in

Table 1. Evaluation index, evaluation grade criteria, and index weight of the water resources carrying capacity.

WRCC Subsystem System	Evaluation Index	Weight	Grade 1 (Loadable)	Grade 2 (Critically Overloaded)	Grade 3 (Overloaded)
Water resources carrying support force subsystem (B1)	C1—per capita water resources (m3/person)	0.1332	≥1670	1670–1000	<1000
	C2—annual water production modulus (104 m3/km2)	0.1332	≥80	80–50	<50
	C3—per capita water supply (m3)	0.1056	≥450	450–350	<350
	C4—vegetation coverage ratio (%)	0.0280	≥40	40–25	<25
Water resources carrying regulation force subsystem (B2)	C5—utilization ratio of water resources (%)	0.0925	≤40	40–70	>70
	C6—per capita GDP (yuan)	0.0769	≥30,000	30,000–8000	<8000
	C7—water consumption ratio of the ecological environment (%)	0.0305	≥5	5–1	<1
Water resources carrying pressure force subsystem (B3)	C8—per capita daily domestic water consumption (L)	0.0369	≤70	70–180	>180
	C9—water consumption per CNY 104 GDP (m3)	0.0792	≤100	100–400	>400
	C10—water consumption per 104 yuan of value-added by industry (m3)	0.0596	≤50	50–200	>200
	C11—population density (person/km2)	0.0792	≤200	200–500	>500
	C12—urbanization ratio (%)	0.0632	≤50	50–80	>80
	C13—farmland irrigation quota (m3/hm2)	0.0792	≤3750	3750–6000	>6000

.

The sample data for the WRCC in Huaibei Plain (2011–2019) were obtained from the Statistical Yearbook of Anhui Province and the Water Resources Bulletin of Anhui Province, etc. Combined with Table 1, firstly, the values of each evaluation index were brought into Equations (2)–(7) to obtain the connection number of a single index of the WRCC evaluation samples. It should be noted that there was a microscopic movement of the connection number components in the above macroscopic state, and the components can be migrated and transformed into each other. Therefore, it was necessary to consider the microscopic movement of the connection number components in order to realize the certainty and accuracy of the evaluation results of the WRCC. In order to quantitatively characterize this microscopic movement, this paper proposed a mobility matrix to solve this challenge. The components v_ijk of Equation (7) were brought into Equation (10) and normalized through Equation (11) to obtain the corrected connection number components $v_{ijk}^{\prime }$ of the single index, and considering the limited space, the changes in the connection number components of the three indexes of Huainan City before and after the transformation through the mobility matrix from 2011 to 2019 were listed tentatively, as shown in

Table 2. Changes in the connection number components of the C₄, C₆, and C₈ evaluation indexes of the water resources carrying capacity in Huainan City.

District	Year	Connection Number Components of Index C4			Connection Number Components of Index C6			Connection Number Componentsof Index C8
		vijk1	vijk2	vijk3	vijk1	vijk2	vijk3	vijk1	vijk2	vijk3
Huainan City	2011	0	0.426	0.574	0.503	0.497	0	0.263	0.500	0.238
	2012	0	0.429	0.571	0.523	0.477	0	0.217	0.500	0.283
	2013	0	0.438	0.562	0.533	0.467	0	0.226	0.500	0.274
	2014	0	0.273	0.727	0.522	0.478	0	0.226	0.500	0.274
	2015	0	0.258	0.742	0.418	0.500	0.082	0.404	0.500	0.096
	2016	0	0.277	0.723	0.454	0.500	0.046	0.267	0.500	0.233
	2017	0	0.360	0.640	0.503	0.497	0	0.268	0.500	0.232
	2018	0	0.329	0.671	0.516	0.484	0	0.265	0.500	0.235
	2019	0	0.301	0.699	0.549	0.451	0	0.194	0.500	0.306
	Year	Corrected Connection Number Components of Index C4			Corrected Connection Number Components of Index C6			Corrected Connection Number Components of Index C8
		${\mathit{v}}_{\mathit{i}\mathit{j}\mathit{k}\mathbf{1}}^{\mathbf{\prime }}$	${\mathit{v}}_{\mathit{i}\mathit{j}\mathit{k}\mathbf{2}}^{\mathbf{\prime }}$	${\mathit{v}}_{\mathit{i}\mathit{j}\mathit{k}\mathbf{3}}^{\mathbf{\prime }}$	${\mathit{v}}_{\mathit{i}\mathit{j}\mathit{k}\mathbf{1}}^{\mathbf{\prime }}$	${\mathit{v}}_{\mathit{i}\mathit{j}\mathit{k}\mathbf{2}}^{\mathbf{\prime }}$	${\mathit{v}}_{\mathit{i}\mathit{j}\mathit{k}\mathbf{3}}^{\mathbf{\prime }}$	${\mathit{v}}_{\mathit{i}\mathit{j}\mathit{k}\mathbf{1}}^{\mathbf{\prime }}$	${\mathit{v}}_{\mathit{i}\mathit{j}\mathit{k}\mathbf{2}}^{\mathbf{\prime }}$	${\mathit{v}}_{\mathit{i}\mathit{j}\mathit{k}\mathbf{3}}^{\mathbf{\prime }}$
	2011	0	0.451	0.549	0.502	0.498	0	0.276	0.469	0.255
	2012	0	0.452	0.548	0.515	0.485	0	0.238	0.469	0.293
	2013	0	0.459	0.541	0.522	0.478	0	0.246	0.469	0.285
	2014	0	0.338	0.662	0.515	0.485	0	0.246	0.469	0.285
	2015	0	0.325	0.675	0.408	0.481	0.111	0.395	0.479	0.126
	2016	0	0.340	0.660	0.445	0.488	0.067	0.279	0.469	0.252
	2017	0	0.404	0.596	0.502	0.498	0	0.280	0.469	0.251
	2018	0	0.381	0.619	0.511	0.489	0	0.278	0.469	0.253
	2019	0	0.360	0.640	0.533	0.467	0	0.219	0.470	0.311

, and then the corrected connection number components of the single index were calculated using Equation (12) to obtain the connection number of the evaluation samples. Finally, the WRCC grade of Huaibei Plain was determined using the level eigenvalue method and the attribute identification method, and these results are shown in

Table 3. Evaluation grade of the WRCC in Huaibei Plain, 2011–2019.

District	Year	Connection Number Components			Level Eigenvalue Method	Attribute Identification Method
		${\mathit{v}}_{\mathit{i}1}^{\prime }$	${\mathit{v}}_{\mathit{i}2}^{\prime }$	${\mathit{v}}_{\mathit{i}3}^{\prime }$
Huaibei City	2011	0.201	0.391	0.409	2.208	3
	2012	0.212	0.397	0.391	2.178	2
	2013	0.213	0.399	0.388	2.175	2
	2014	0.216	0.402	0.382	2.166	2
	2015	0.216	0.393	0.391	2.175	2
	2016	0.241	0.398	0.361	2.120	2
	2017	0.250	0.394	0.356	2.107	2
	2018	0.279	0.415	0.306	2.027	2
	2019	0.239	0.355	0.406	2.167	3
Bozhou City	2011	0.179	0.410	0.411	2.232	3
	2012	0.217	0.407	0.376	2.159	2
	2013	0.210	0.398	0.391	2.181	2
	2014	0.241	0.415	0.343	2.102	2
	2015	0.226	0.410	0.364	2.138	2
	2016	0.246	0.429	0.325	2.080	2
	2017	0.265	0.429	0.306	2.041	2
	2018	0.270	0.422	0.308	2.038	2
	2019	0.220	0.389	0.391	2.171	2
Suzhou City	2011	0.225	0.404	0.371	2.145	2
	2012	0.220	0.400	0.380	2.160	2
	2013	0.227	0.402	0.371	2.145	2
	2014	0.246	0.404	0.35	2.104	2
	2015	0.244	0.398	0.358	2.113	2
	2016	0.261	0.417	0.323	2.062	2
	2017	0.275	0.415	0.310	2.035	2
	2018	0.290	0.429	0.281	1.990	2
	2019	0.264	0.378	0.357	2.093	2
Bengbu City	2011	0.187	0.465	0.347	2.160	2
	2012	0.196	0.448	0.356	2.160	2
	2013	0.212	0.446	0.342	2.130	2
	2014	0.216	0.468	0.316	2.100	2
	2015	0.213	0.464	0.323	2.111	2
	2016	0.238	0.465	0.297	2.058	2
	2017	0.247	0.464	0.289	2.041	2
	2018	0.278	0.468	0.253	1.975	2
	2019	0.242	0.403	0.355	2.113	2
Fuyang City	2011	0.144	0.399	0.457	2.312	3
	2012	0.157	0.399	0.444	2.287	3
	2013	0.161	0.407	0.432	2.271	3
	2014	0.211	0.422	0.367	2.156	2
	2015	0.195	0.417	0.388	2.192	2
	2016	0.219	0.437	0.343	2.124	2
	2017	0.242	0.446	0.313	2.071	2
	2018	0.237	0.431	0.332	2.096	2
	2019	0.212	0.391	0.397	2.184	2
Huainan City	2011	0.171	0.385	0.444	2.272	3
	2012	0.16	0.366	0.474	2.314	3
	2013	0.178	0.375	0.447	2.269	3
	2014	0.181	0.413	0.406	2.225	3
	2015	0.178	0.402	0.419	2.241	3
	2016	0.243	0.417	0.340	2.097	2
	2017	0.267	0.422	0.310	2.043	2
	2018	0.276	0.425	0.298	2.022	2
	2019	0.205	0.375	0.420	2.215	3

. Meanwhile, based on the level eigenvalue value, the spatial and temporal distributions of the WRCC in Huaibei Plain was drawn, as shown in Figure 5.

According to Table 2, in 2011–2019, the similarity degree component of C₄ was zero; the opposition degree component was the largest, the opposition degree component after the transformation of the mobility matrix decreased, the difference degree component increased, and the similarity degree component was still zero. This was because the similarity degree component in the original connection number components was zero, and there was no increment in the direction of the similarity degree in the process of the migration movement of each component, i.e., a = 0. In 2015, the original connection number components of C₆ were 0.418, 0.500, and 0.082, and 0.408, 0.481, and 0.111 after migration transformation, and the original connection number components of C₈ in 2012 were 0.217, 0.500, and 0.283, and 0.238, 0.469, and 0.293 after transformation. From these results, it can be interpreted that the difference degree component in the original connection number components was the largest, which was reduced after the migration transformation, and the sum of the similarity degree and the opposite degree components increased, which may be that both the opposition degree and similarity degree components increased at the same time, such as the C₈ in 2012, or one of them increased and the other decreased, such as the index C₆ in 2015.

The results of the microscopic movement between the connection number components further indicate that the maximum value in the ternary connection number component will decrease in the value of that component after the microscopic migration movement. According to the fact that the sum of the similarity, difference, and opposition degree components is one, the sum of the other two components will increase, and the change in both of them is related to the size of the two components themselves, the ratio of the increment expressed by the semi-partial connection number to the original connection number components, and the size of the original connection number components that are being migrated.

At present, some studies have neglected the microscopic movement among the connection number components [46,47], resulting in unreliable and unstable evaluation results, which are not in accordance with the actual water resources carrying states of the region and cannot provide an effective decision-making basis for the management of regional water resources. In this paper, the migration and transformation of the connection number components under the macro-static and micro-movement states were comprehensively considered, and the mobility matrix was proposed based on the partial connection number, which can realize the qualitative and quantitative portrayals of the direction and incremental size of the movement of the connection number components at the micro-level under the macro-static state. Taking the connection number component as the driving object, taking the “information energy” stored in the connection number component as the driving force, and taking the migration to different opposing levels as the driving direction, further data mining and information utilization can be carried out on the connection number information of the WRCC evaluation index value and grade, and fully analyze the connection number information and its uncertainty, so as to determine the connection number in a more reasonable way and deeply understand the micro-movement law among the connection number components, and at the same time, to reduce the uncertainty of the WRCC system, which provides a new viewpoint for the accurate evaluation of the WRCC.

According to Table 3 and Figure 5:

(1)

From 2011 to 2019, the WRCC grades of Huaibei City, Bozhou City, Fuyang City, and Huainan City were 2.027–2.208, 2.038–2.232, 2.071–2.312, and 2.022–2.314, respectively, which were between critically overloaded and overloaded states; those of Suzhou City and Bengbu City, 1.990–2.160 and 1.975–2.160, respectively, were between loadable and critically overloaded states in 2018, and between critically overloaded and overloaded states in the remaining years. Overall, the WRCC of Huaibei Plain fluctuated around grade 2 from 2011 to 2019; the evaluation grades calculated using the attribute identification method were either grade 2 or grade 3. The grades of the two methods were in good agreement, indicating that the mobility matrix proposed in this paper, which quantitatively portrayed the microscopic movements among the connection number components, were feasible and effective, verifying the reliability of the proposed method. It provides a new insight into the accurate and reasonable determination of the connection number and the quantitative evaluation of the WRCC, and will provide technical support for regional water resources management with the goal of achieving sustainable utilization.

(2)

In time, the WRCC in Huaibei Plain as a whole showed a good development trend gradually, e.g., Bozhou City changed from 2.232 in 2011 to 2.038 in 2018 and to 2.171 in 2019. It is associated with the strictest water resource management system in Anhui Province since 2013. In space, the WRCC of Fuyang City and Huainan City was worse than that of the other four cities from 2011 to 2019. Over the nine years, the average grades for Fuyang City and Huainan City were 2.26 and 2.43, while the grades for Huaibei City, Bozhou City, Suzhou City, and Bengbu City were 2.19, 2.12, 2.05, and 2.05, respectively, which was mainly related to the water resources, the number of resident population, GDP, and the farmland irrigation quota; the WRCC of Huainan City was the poorest among the six cities in 2013–2015 and 2019, and Fuyang City was the poorest among the six cities in 2011, 2013, and 2016. These findings indicate that the method of this paper, in time, realized the dynamic analysis of the WRCC system and a judgment of its development trend; in space, the spatial difference in the WRCC in Huaibei Plain can be accurately identified. This will help to quantitatively and qualitatively analyze the WRCC system and grasp the development trend of the water resources system.

(3)

The WRCC in Huaibei Plain had a tendency to become worse during 2019 due to the drought in Anhui Province. By reviewing the available information, from March 2019, rainfall continued to be low, with 70% less rainfall than normal during the plum rainy period in Huaibei. From August 12th to November 11th, the weather in Anhui Province was sunny and hot with little rain, with the province’s average rainfall being 70% less than normal. In 2019, the annual rainfall in Huaibei City, Bozhou City, Suzhou City, Bengbu City, Fuyang City, and Huainan City was 611.7 mm, 566.8 mm, 593.4 mm, 614.9 mm, 552.1 mm, and 560.5 mm, accounting for 65.37%, 60.57%, 63.41%, 65.71%, 59.00%, and 59.90% of the province’s annual rainfall (935.8 mm), respectively. Soil moisture monitoring results showed that the area of Anhui Province undergoing severe drought was 2.0 × 10⁴ km², moderate drought was about 4.3 × 10⁴ km², and mild drought was about 4.7 × 10⁴ km², with the drought area of soil moisture accounting for about 79% of the total area of the province. The affected area in the province was 6793.3 km², of which 1346.7 km² was along the northern part of the Huaihe River. In addition, the per capita water resources in Huaibei Plain also decreased in 2019 compared to previous years, with the per capita water resources of Huaibei City, Bozhou City, Suzhou City, Bengbu City, Fuyang City, and Huainan City being 211.45 m³, 279.16 m³, 312.84 m³, 311.86 m³, 207.05 m³, and 222.39 m³, respectively, accounting for 24.93%, 32.92%, 36.89%, 36.77%, 24.41%, and 26.22% of the province’s per capita water resources (848.07 m³), respectively. From the perspective of the physical causes that form the water resources carrying state, as the factors affecting the water resources carrying state in Huaibei Plain (such as water resources) changed in 2019, its WRCC grade was bound to respond under the effect of a certain relationship’s structure; the response results were consistent with actual perceptions, meaning that the evaluation method constructed in this paper was more responsive under the changing conditions. This will provide the basis for managers to establish more targeted water management measures, as a more responsive approach has greater precision and more comprehensive information.

(4)

Cui et al. [22] constructed a set pair analysis evaluation model to obtain a WRCC grade value of 2.32 for Huaibei City, which was between critically overloaded and overloaded states. Gao et al. [48] reported that a fuzzy integrated model was used to estimate the WRCC in Suzhou City, which was close to being overloaded and had little potential for utilization of water resources. Wang [49] used a multi-level fuzzy-integrated evaluation model based on AHP assignment, and their evaluation results showed the largest affiliation to the overloaded state, indicating that the overall WRCC of Fuyang City is weak. Furthermore, Bai [50] used a coupled evaluation model based on gray correlation, and SPA calculated the WRCC grade of Bozhou City from 2012 to 2015, all of which were grade 2; Fuyang City was grade 3 from 2012 to 2013 and grade 2 from 2014 to 2015; Huainan City was grade 3 in 2012, 2013, and 2015 and grade 2 in 2014. The results of this paper were well consistent with available studies, all suggesting that the WRCC in Huaibei Plain is poor.

4.2. Diagnosis of the Water Resources Carrying Capacity Obstacle factors

Therefore, further diagnosis of the obstacle factors of the WRCC system in Huaibei Plain was required. Firstly, the dynamic difference degree coefficient, determined using Equation (16), was brought into the single index connection number (Equation (11)) to compute the connection number value interval whose arithmetic mean was regarded as the set pair potential value of index j; then, Equation (17) was applied to make linear changes to the connection number value; finally, the distance between each index value and the ideal solutions was computed using Equations (18)–(20) and the obstacle degree, respectively. Apparently, the more the obstacle degree value tended to one, the more the index was the obstacle factor of the WRCC system. These results are shown in

Table 4. Obstacle degree of the WRCC in Huaibei Plain, 2011–2019.

District	Index	2011	2012	2013	2014	2015	2016	2017	2018	2019
Huaibei City	C1	0.9683	0.9558	0.9599	0.9529	0.9609	0.9465	0.9437	0.9064	0.9759
	C2	0.9286	0.9158	0.9193	0.9117	0.9195	0.9043	0.9005	0.8623	0.9328
	C3	0.8991	0.9036	0.8985	0.9008	0.9061	0.9092	0.9120	0.9142	0.9176
	C4	0.8919	0.8899	0.8809	0.8841	0.8814	0.8806	0.8840	0.8822	0.8825
	C5	0.8702	0.8612	0.8639	0.8600	0.8618	0.5837	0.5072	0.1254	0.8661
	C6	0.3001	0.1775	0.1342	0.1290	0.1296	0.1264	0.1134	0.1071	0.0963
	C7	0.9268	0.9224	0.8764	0.8505	0.8334	0.6096	0.6392	0.5346	0.3081
	C8	0.3964	0.3800	0.3821	0.3704	0.3724	0.3927	0.3943	0.3876	0.5048
	C9	0.1334	0.1194	0.1128	0.1055	0.1017	0.0967	0.0858	0.0809	0.0734
	C10	0.1186	0.1064	0.1080	0.1085	0.1092	0.1088	0.1008	0.0986	0.1053
	C11	0.8885	0.8877	0.8862	0.8865	0.8870	0.8870	0.8871	0.8875	0.8911
	C12	0.3135	0.3478	0.3756	0.4020	0.4208	0.4469	0.4744	0.5020	0.5162
	C13	0.0490	0.0383	0.0441	0.0391	0.0386	0.0346	0.0339	0.0327	0.0161
Bozhou City	C1	0.9351	0.9269	0.9354	0.9115	0.9282	0.9139	0.9066	0.9165	0.9602
	C2	0.9161	0.9087	0.9150	0.8925	0.9072	0.8936	0.8856	0.8936	0.9330
	C3	0.9204	0.9351	0.9287	0.9339	0.9283	0.9069	0.9095	0.9113	0.9102
	C4	0.8864	0.8854	0.8832	0.8914	0.8885	0.8864	0.8871	0.8873	0.8881
	C5	0.5184	0.2313	0.4506	0.1310	0.3396	0.1395	0.1298	0.1381	0.8603
	C6	0.6648	0.6130	0.5745	0.5309	0.5057	0.4595	0.4135	0.3530	0.1335
	C7	0.9167	0.9061	0.9011	0.8854	0.8908	0.6656	0.4288	0.3980	0.3920
	C8	0.2794	0.2815	0.3017	0.3071	0.3083	0.3052	0.3012	0.2934	0.8636
	C9	0.3514	0.2412	0.2447	0.1792	0.1803	0.1482	0.1335	0.1245	0.1023
	C10	0.6069	0.5051	0.4685	0.3279	0.3150	0.3244	0.1623	0.1661	0.2676
	C11	0.8795	0.8805	0.8829	0.8831	0.8832	0.8847	0.8851	0.8859	0.8706
	C12	0.1043	0.1081	0.1112	0.1139	0.1166	0.1192	0.1222	0.1246	0.1269
	C13	0.0782	0.0456	0.0325	0.0316	0.0446	0.0427	0.0333	0.0306	0.0252
Suzhou City	C1	0.9216	0.9288	0.9319	0.9263	0.9347	0.9188	0.9148	0.8732	0.9529
	C2	0.9078	0.9139	0.9161	0.9105	0.9174	0.9029	0.8982	0.8601	0.9312
	C3	0.9266	0.9262	0.9236	0.9298	0.9337	0.9192	0.9229	0.9234	0.9212
	C4	0.7031	0.6790	0.6606	0.8090	0.7847	0.8174	0.7273	0.7545	0.7766
	C5	0.2102	0.3294	0.3795	0.1987	0.2868	0.1331	0.1256	0.0895	0.5376
	C6	0.6042	0.5496	0.5058	0.4522	0.4127	0.3611	0.3057	0.2022	0.1302
	C7	0.9641	0.9641	0.9315	0.9139	0.9112	0.7030	0.6265	0.5946	0.5069
	C8	0.4062	0.3993	0.3937	0.3962	0.3986	0.3954	0.3894	0.3949	0.4892
	C9	0.2582	0.2118	0.1778	0.1327	0.1235	0.1179	0.1098	0.1019	0.0906
	C10	0.4314	0.4985	0.3842	0.3169	0.2764	0.2384	0.1369	0.1360	0.2243
	C11	0.8739	0.8737	0.8727	0.8728	0.8735	0.8739	0.8740	0.8741	0.8661
	C12	0.1083	0.1120	0.1150	0.1175	0.1201	0.1227	0.1256	0.1279	0.1301
	C13	0.0313	0.0331	0.0322	0.0223	0.0273	0.0292	0.0250	0.0216	0.0065
Bengbu City	C1	0.8932	0.9212	0.9286	0.8968	0.8993	0.8967	0.8912	0.8659	0.9532
	C2	0.8856	0.9094	0.9154	0.8865	0.8883	0.8852	0.8788	0.7711	0.9325
	C3	0.2997	0.4003	0.3136	0.8605	0.5403	0.1392	0.2927	0.4618	0.4031
	C4	0.9055	0.9036	0.8832	0.8897	0.8871	0.8908	0.8907	0.8883	0.8893
	C5	0.8605	0.8706	0.8757	0.5974	0.8597	0.8514	0.6353	0.2618	0.8851
	C6	0.3516	0.2346	0.1375	0.1285	0.1220	0.1128	0.1006	0.0865	0.0474
	C7	0.7086	0.6328	0.6522	0.6050	0.6187	0.7322	0.7640	0.7322	0.4779
	C8	0.3981	0.4425	0.4400	0.4420	0.4403	0.4291	0.4255	0.4271	0.8757
	C9	0.4114	0.3490	0.3129	0.1838	0.2055	0.1762	0.1359	0.1231	0.1116
	C10	0.4483	0.2906	0.2018	0.1319	0.1281	0.1129	0.1244	0.1101	0.1218
	C11	0.8693	0.8691	0.8689	0.8696	0.8704	0.8709	0.8711	0.8715	0.8651
	C12	0.1348	0.1378	0.1401	0.1748	0.2191	0.2635	0.3041	0.3483	0.3772
	C13	0.4724	0.4664	0.4620	0.1311	0.3044	0.3750	0.3077	0.2082	0.1215
Fuyang City	C1	0.9627	0.9565	0.9506	0.9259	0.9386	0.9205	0.9054	0.9273	0.9769
	C2	0.9231	0.9168	0.9108	0.8849	0.8971	0.8784	0.8618	0.8827	0.9336
	C3	0.9225	0.9328	0.9284	0.9350	0.9280	0.9037	0.9045	0.9067	0.9066
	C4	0.8912	0.8915	0.8856	0.8843	0.8830	0.8828	0.8860	0.8844	0.8841
	C5	0.8661	0.8599	0.8456	0.2858	0.5313	0.2824	0.1331	0.3427	0.8715
	C6	0.7198	0.6725	0.6357	0.5949	0.5732	0.5341	0.4866	0.4344	0.1345
	C7	0.8908	0.8897	0.8875	0.8698	0.8707	0.7241	0.3889	0.3828	0.3688
	C8	0.3322	0.3396	0.3510	0.3515	0.3581	0.3454	0.3401	0.3319	0.6973
	C9	0.4277	0.3392	0.3263	0.2538	0.2582	0.2226	0.1824	0.1391	0.1062
	C10	0.4714	0.4293	0.3702	0.2521	0.2058	0.1805	0.1923	0.1393	0.1952
	C11	0.9179	0.9188	0.9208	0.9205	0.9192	0.9220	0.9233	0.9234	0.8917
	C12	0.1088	0.1122	0.1150	0.1177	0.1203	0.1231	0.1260	0.1289	0.1313
	C13	0.1182	0.0978	0.1023	0.0908	0.1024	0.1007	0.0929	0.0883	0.0774
Huainan City	C1	0.9565	0.9641	0.9646	0.9506	0.9328	0.9058	0.9013	0.8891	0.9733
	C2	0.9075	0.9154	0.9156	0.8999	0.8811	0.2253	0.1375	0.1228	0.8994
	C3	0.0567	0.0662	0.0764	0.1000	0.4734	0.0945	0.0942	0.1023	0.1098
	C4	0.8830	0.8823	0.8793	0.9291	0.9336	0.9280	0.9031	0.9126	0.9207
	C5	0.9782	0.9979	0.9885	0.9206	0.8958	0.8755	0.8731	0.8658	0.9726
	C6	0.1398	0.1331	0.1299	0.1334	0.2942	0.2350	0.1395	0.1353	0.1247
	C7	0.8726	0.8602	0.8475	0.8120	0.7823	0.6690	0.5641	0.5415	0.5096
	C8	0.4862	0.5370	0.5262	0.5269	0.3143	0.4814	0.4799	0.4835	0.5624
	C9	0.5304	0.4671	0.4297	0.3975	0.3483	0.4706	0.4363	0.3914	0.3217
	C10	0.9002	0.8783	0.8688	0.8733	0.8731	0.8721	0.6426	0.6942	0.8616
	C11	0.6816	0.9029	0.9027	0.9027	0.9952	0.9988	0.9991	0.9991	0.9677
	C12	0.4761	0.5055	0.5313	0.5537	0.4190	0.4454	0.4716	0.4836	0.5007
	C13	0.8600	0.8599	0.5274	0.3921	0.0856	0.3871	0.3803	0.2596	0.1403

Notes: C₁ = per capita water resources, C₂ = annual water production modulus, C₃ = per capita water supply, C₄ = vegetation coverage ratio, C₅ = utilization ratio of water resources, C₆ = per capita GDP, C₇ = water consumption ratio of the ecological environment, C₈ = per capita daily domestic water consumption, C₉ = water consumption per CNY 10⁴ GDP, C₁₀ = water consumption per 10⁴ yuan value-added by industry, C₁₁ = population density, C₁₂ = urbanization ratio, and C₁₃ = farmland irrigation quota.

(the bold data in the table indicate the top five obstacle degree values, and the corresponding indexes were the key obstacle factors).

To reflect the obstacle degree of the indexes more clearly and intuitively in Huaibei Plain, Table 4 was drawn as a broken line chart, as shown in Figure 6. Simultaneously, the key obstacles to the improvement in the WRCC can be obtained by ranking the obstacles in descending order. Here, the top five were selected; these are shown in Table 4.

According to Figure 6 and

Table 5. Key obstacle factors of WRCC in Huaibei Plain, 2011–2019.

District	Year	Item	The First Obstacle Factor	The Second Obstacle Factor	The Third Obstacle Factor	The Fourth Obstacle Factor	The Fifth Obstacle Factor
Huaibei City	2011	Index/obstacle degree	C1/0.9683	C2/0.9286	C7/0.9268	C3/0.8991	C4/0.8919
	2012	Index/obstacle degree	C1/0.9558	C7/0.9224	C2/0.9158	C3/0.9036	C4/0.8899
	2013	Index/obstacle degree	C1/0.9599	C2/0.9193	C3/0.8985	C11/0.8862	C4/0.8809
	2014	Index/obstacle degree	C1/0.9529	C2/0.9117	C3/0.9008	C11/0.8865	C4/0.8841
	2015	Index/obstacle degree	C1/0.9606	C2/0.9195	C3/0.9061	C11/0.8870	C4/0.8806
	2016	Index/obstacle degree	C1/0.9465	C3/0.9092	C2/0.9043	C11/0.8870	C4/0.8806
	2017	Index/obstacle degree	C1/0.9437	C3/0.9120	C2/0.9005	C11/0.8871	C4/0.8840
	2018	Index/obstacle degree	C3/0.9142	C1/0.9064	C11/0.8875	C4/0.8822	C2/0.8623
	2019	Index/obstacle degree	C1/0.9759	C2/0.9328	C3/0.9176	C11/0.8911	C4/0.8825
Bozhou City	2011	Index/obstacle degree	C1/0.9351	C3/0.9204	C7/0.9167	C2/0.9161	C4/0.8864
	2012	Index/obstacle degree	C3/0.9351	C1/0.9269	C2/0.9087	C7/0.9061	C4/0.8854
	2013	Index/obstacle degree	C1/0.9354	C3/0.9287	C2/0.9150	C7/0.9011	C4/0.8832
	2014	Index/obstacle degree	C3/0.9339	C1/0.9115	C2/0.8925	C4/0.8914	C7/0.8854
	2015	Index/obstacle degree	C3/0.9283	C1/0.9282	C2/0.9072	C7/0.8908	C4/0.8885
	2016	Index/obstacle degree	C1/0.9139	C3/0.9069	C2/0.8936	C4/0.8864	C11/0.8847
	2017	Index/obstacle degree	C3/0.9095	C1/0.9066	C4/0.8871	C2/0.8856	C11/0.8851
	2018	Index/obstacle degree	C1/0.9165	C3/0.9113	C2/0.8936	C4/0.8873	C11/0.8859
	2019	Index/obstacle degree	C1/0.9602	C2/0.9330	C3/0.9102	C4/0.8881	C11/0.8706
Suzhou City	2011	Index/obstacle degree	C7/0.9641	C3/0.9266	C1/0.9216	C2/0.9078	C11/0.8739
	2012	Index/obstacle degree	C7/0.9641	C1/0.9288	C3/0.9262	C2/0.9139	C11/0.8727
	2013	Index/obstacle degree	C1/0.9319	C7/0.9315	C3/0.9236	C2/0.9161	C11/0.8727
	2014	Index/obstacle degree	C3/0.9298	C1/0.9263	C7/0.9139	C2/0.9105	C11/0.8728
	2015	Index/obstacle degree	C1/0.9347	C3/0.9337	C2/0.9174	C7/0.9112	C11/0.8735
	2016	Index/obstacle degree	C3/0.9192	C1/0.9188	C2/0.9029	C11/0.8739	C4/0.8174
	2017	Index/obstacle degree	C3/0.9229	C1/0.9148	C2/0.8982	C11/0.8740	C4/0.7273
	2018	Index/obstacle degree	C3/0.9234	C11/0.8741	C1/0.8732	C2/0.8601	C4/0.7545
	2019	Index/obstacle degree	C1/0.9529	C2/0.9312	C3/0.9212	C11/0.8661	C4/0.7766
Bengbu City	2011	Index/obstacle degree	C4/0.9055	C1/0.8932	C2/0.8856	C11/0.8693	C5/0.8605
	2012	Index/obstacle degree	C1/0.9212	C2/0.9094	C4/0.9036	C5/0.8706	C11/0.8691
	2013	Index/obstacle degree	C1/0.9286	C2/0.9154	C4/0.8832	C5/0.8757	C11/0.8689
	2014	Index/obstacle degree	C1/0.8968	C4/0.8897	C2/0.8865	C11/0.8696	C3/0.8605
	2015	Index/obstacle degree	C1/0.8993	C2/0.8883	C4/0.8871	C11/0.8704	C5/0.8597
	2016	Index/obstacle degree	C1/0.8967	C4/0.8908	C2/0.8852	C11/0.8709	C5/0.8514
	2017	Index/obstacle degree	C1/0.8912	C4/0.8907	C2/0.8788	C11/0.8711	C7/0.7674
	2018	Index/obstacle degree	C4/0.8883	C11/0.8715	C1/0.8659	C2/0.7711	C7/0.7322
	2019	Index/obstacle degree	C1/0.9532	C2/0.9325	C4/0.8893	C5/0.8851	C8/0.8757
Fuyang City	2011	Index/obstacle degree	C1/0.9627	C2/0.9231	C3/0.9225	C11/0.9179	C4/0.8912
	2012	Index/obstacle degree	C1/0.9565	C3/0.9328	C11/0.9188	C2/0.9168	C4/0.8915
	2013	Index/obstacle degree	C1/0.9506	C3/0.9284	C11/0.9208	C2/0.9108	C7/0.8875
	2014	Index/obstacle degree	C3/0.9350	C1/0.9259	C11/0.9205	C2/0.8849	C4/0.8843
	2015	Index/obstacle degree	C1/0.9386	C3/0.9280	C11/0.9192	C2/0.8971	C4/0.8830
	2016	Index/obstacle degree	C11/0.9220	C1/0.9205	C3/0.9037	C4/0.8828	C2/0.8784
	2017	Index/obstacle degree	C11/0.9233	C1/0.9054	C3/0.9045	C4/0.8860	C2/0.8618
	2018	Index/obstacle degree	C1/0.9273	C11/0.9234	C3/0.9067	C4/0.8844	C2/0.8827
	2019	Index/obstacle degree	C1/0.9769	C2/0.9336	C3/0.9066	C11/0.8917	C4/0.8841
Huainan City	2011	Index/obstacle degree	C5/0.9782	C1/0.9565	C2/0.9075	C10/0.9002	C4/0.8830
	2012	Index/obstacle degree	C5/0.9979	C1/0.9641	C2/0.9154	C11/0.9029	C4/0.8823
	2013	Index/obstacle degree	C5/0.9885	C1/0.9646	C2/0.9156	C11/0.9027	C4/0.8793
	2014	Index/obstacle degree	C1/0.9506	C4/0.9291	C5/0.9206	C11/0.9027	C2/0.8999
	2015	Index/obstacle degree	C11/0.9952	C4/0.9336	C1/0.9328	C5/0. 8958	C2/0.8811
	2016	Index/obstacle degree	C11/0.9988	C4/0.9280	C1/0.9058	C5/0.8755	C10/0.8721
	2017	Index/obstacle degree	C11/0.9991	C4/0.9031	C1/0.9013	C5/0.8731	C10/0.6426
	2018	Index/obstacle degree	C11/0.9991	C4/0.9126	C1/0.8891	C5/0.8658	C10/0.6942
	2019	Index/obstacle degree	C1/0.9733	C5/0.9726	C11/0.9677	C4/0.9207	C2/0.8994

:

(1)

The WRCC obstacle degree of each index changed with the evaluation area and the evaluation year, and even for the same index, its obstacle degree changed with the evaluation sample, which showed that the method constructed in this paper can dynamically diagnose and identify obstacle factors. Taking Huainan City as an example, the obstacle degree was ranked from largest to smallest as C₁₁ > C₄ > C₁ > C₅ > C₁₀ > C₇ > C₈ > C₁₂ > C₉ > C₁₃ > C₆ > C₂ > C₃ in 2018, and C₁ > C₅ > C₁₁ > C₄ > C₂ > C₁₀ > C₈ > C₇ > C₁₂ > C₉ > C₁₃ > C₆ > C₃ in 2019, respectively. In terms of the evaluation indexes, the water resources carrying capacity state of six cities in the Huaibei Plain developed positively from 2011–2018, while the main reason for the deterioration in 2019 was attributed to the low precipitation in that year, the decrease in water resources, and the high degree of water resource utilization. The average obstacle degree of the per capita water resources C₁ (average value of 2011–2018) of the six cities in the Huaibei Plain increased from 0.9263 to 0.9654 in 2019; the average obstacle degree of the annual water production modulus C₂ increased from 0.8517 to 0.9271, the average obstacle degree of the utilization ratio of water resources C₅ increased from 0.5567 to 0.8322, and the average obstacle degree of the per capita daily domestic water consumption increased from 0.3887 to 0.6655. These data showed that the precipitation in the cities of Huaibei Plain in the year 2019 ranged from 552.1 mm–614.9 mm, which was about 40% less compared to 2018 and about 30% less compared to the multi-year average. It can be seen that the quantitative diagnostic method of the regional WRCC obstacle factors based on the CN and TOPSIS constructed in this paper can effectively identify the change process of carrying capacity evaluation indexes, and the year-by-year change in the obstacle degree can quantitatively analyze the reasons for the change in the carrying state.

(2)

The main obstacle factors affecting the enhancement in the WRCC in Huaibei City were C₁, C₂, C₃, C₄, and C₇ in 2011–2012, and C₁, C₂, C₃, C₄, and C₁₁ in 2013–2019.The water consumption ratio of the ecological environment C₇, has been improved, and its obstacle degree decreased from 0.9268 in 2011 to 0.3081 in 2019, and was no longer a key obstacle factor. From this, it can be determined that the key obstacle factors hindering the improvement in the WRCC in Huaibei City were the per capita water resources C₁, the annual water production modulus C₂, the per capita water supply C₃, the vegetation coverage ratio C₄, and the population density C₁₁, of which there are four (4/4 × 100 = 100%), zero, and one (1/6 × 100% = 16.7%) in the support, regulation, and pressure force subsystems, respectively. At the same time, the per capita water resources C₁, as the first obstacle factor, appeared eight times (years) in nine years, indicating that the main reason for the WRCC of Huaibei City to be between the critically overloaded and overloaded states for a long time was the insufficient support force. Based on the year-by-year changes in the obstacle degree of the indexes to analyze the causal mechanism of the carrying state, the obstacle degree of the utilization ratio of water resources C₅, the per capita GDP C₆, the water consumption ratio of the ecological environment C₇, the water consumption per CNY 10⁴ GDP C₉, the water consumption per 10⁴ yuan value-added by industry C₁₀, and the farmland irrigation quota C₁₃ in Huaibei City from 2011 to 2018 had been decreasing gradually, i.e., these indexes improved. In terms of the evaluation indexes, this was also an important reason why the city’s WRCC grade was decreasing from 2.208 in 2011 to 2.027 in 2018, showing an improving trend. It was found that the utilization ratio of water resources in Huaibei City was 98.45% in 2011, showing a downward trend in recent years, which was directly related to the control of the “three red lines” (the red line of control of water resources development and utilization, etc.), reflecting the science and effectiveness of the most stringent water resources management system. The water consumption ratio of the ecological environment increased from 2 × 10⁶ m³ in 2011 to 1.8 × 10⁷ m³ in 2019, and the water consumption ratio of the ecological environment (the proportion of ecological water consumption to total water consumption) increased from 0.39% to 4.27%. The above two indexes, as importants part of the water resources regulation force, improved the water resources carrying state of Huaibei City. In addition, the indexes of water consumption per CNY 10⁴ GDP and water consumption per 10⁴ yuan value-added by industry, which were the pressure forces of the water resources, were also continuing to develop in a favorable way, and the pressure force was reduced. Therefore, in terms of the evaluation subsystems, the WRCC of Huaibei City developed in a positive direction due to the weakness in the pressure force and the enhancement in the regulation force. The main reason for the deterioration in the water resources carrying state in Huaibei City in 2019 was the low precipitation, the decrease in the water resources, the over-exploitation of the water resources, and the increase in per capita daily domestic water consumption, coupled with the fact that this city is located far away from the Huaihe River, the conditions for the diversion and transfer of water are insufficient, and the relevant data showed that the total water resources of Huaibei City was 1.8 × 10⁸ m³ in 2019. which was 63.5% less than that of 2018, and 42.3% less than the average value of many years. The diagnostic results of this city’s water resources carrying state were in line with the actual situation, which indicated that the quantitative diagnostic method of the WRCC factors based on the connection number and TOPSIS, which was constructed in this paper, was effective.

(3)

The main obstacle factors affecting the enhancement in the WRCC in Bozhou City were C₁, C₂, C₃, C₄, and C₇ in 2011–2015, and C₁, C₂, C₃, C₄, and C₁₁ in 2013–2019. The water consumption ratio of the ecological environment C₇ increased from 0.36% in 2011 to 3.76% in 2019, and its obstacle degree declined from 0.9167 to 0.3920, and the index was no longer a key factor restricting the improvement in the WRCC. Therefore, the key obstacle factors hindering the enhancement in its WRCC were the per capita water resources C₁, the annual water production modulus C₂, the per capita water supply C₃, the vegetation coverage ratio C₄, and the population density C₁₁, which was consistent with that of Huaibei City. Insufficient support force was still an important reason for Bozhou City’s WRCC to jump from a critically overloaded or overloaded to a loadable state. In Bozhou City, the obstacle degree of the per capita GDP C₆, the water consumption ratio of the ecological environment C₇, the water consumption per CNY 10⁴ GDP C₉, and the water consumption per 10⁴ yuan value-added by industry C₁₀ gradually decreased from 2011 to 2018, i.e., these indexes improved, while the overall obstacle degree of the per capita daily domestic water consumption C₈ gradually increased, i.e., it deteriorated; the utilization ratio of water resources C₅ fluctuated a lot. In terms of the evaluation indexes, the water resources carrying state of Bozhou City developed to a better state due to the degree in the improvements of C₆, C₇, C₉, and C₁₀ being greater than the degree of deterioration of C₈, indicating that the increase in the per capita GDP, the increase in the water consumption ratio of the ecological environment, and the decrease in water consumption per CNY 10⁴ GDP and water consumption per 10⁴ yuan value-added by industry promoted its WRCC; meanwhile, the increase in the per capita daily domestic water consumption impeded the improvement of its carrying capacity, but the former was greater than the latter, and the overall performance of the WRCC showed a trend of improvement. In terms of the evaluation subsystems, the main reason was the increase in the regulation force. In addition, the C₅ fluctuated greatly, and Bozhou City should earnestly implement the most stringent water resources system so that the utilization of their water resources can be kept in a reasonable range. The deterioration in the water resources carrying state of Bozhou City in 2019 was due to the combined effects of weakening support and regulation force and strengthening pressure force, with the per capita water resources C₁ and the annual water production modulus C₂ obstacle degree in the support force subsystem deteriorating from 0.9165 and 0.8936 in 2018 to 0.9602 and 0.9330 in 2019, respectively, the utilization ratio of water resources C₅ in the regulation force subsystem deteriorating from 0.1381 in 2018 to 0.8603 in 2019, and the per capita daily domestic water consumption C₈ in the pressure force subsystem deteriorating from 0.2934 in 2018 to 0.8636 in 2019.

(4)

The key obstacle factors affecting the enhancement in the WRCC in Suzhou City were C₁, C₂, C₃, C₇, and C₁₁ in 2011–2015, and C₁, C₂, C₃, C₄, and C₁₁ in 2016–2019. The water consumption ratio of the ecological environment C₇ improved, while the vegetation coverage ratio C₄ deteriorated, and the obstacle degree ranged between 0.6606–0.8174, keeping a high level. Therefore, the key obstacle factors hindering the improvement in the WRCC in Suzhou City were the per capita water resources C₁, the annual water production modulus C₂, the per capita water supply C₃, the vegetation coverage ratio C₄, and the population density C₁₁, which showed that the water resources available was less while the population was more, and that the vegetation coverage was less, i.e., insufficient support force constraints on the improvement of the city’s WRCC. According to the related literature [51], the total water resources of Suzhou City were less, which led to its water resources carrying support force always being insufficient, which was consistent with the results of the analysis in this study. The obstacle degree of the utilization ratio of water resources C₅, the per capita GDP C₆, the water consumption ratio of the ecological environment C₇, the water consumption per CNY 10⁴ GDP C₉, the water consumption per 10⁴ yuan value-added by industry C₁₀, and the farmland irrigation quota C₁₃ in Suzhou City from 2011 to 2018 gradually decreased, i.e., these indexes improved, while the obstacle degree of the vegetation coverage ratio C₄ and the urbanization ratio C₁₂ gradually increased, i.e., these indexes worsened. In terms of the evaluation indexes, Suzhou City’s WRCC state developed into a better due to the overall reduction in the utilization ratio of water resources, the improvement in the economic development level, the increase in water consumption in the ecological environment year by year, and the reduction in the farmland irrigation quota. Although C₄ and C₁₂ were gradually deteriorating, and their obstacle degree deteriorated from 0.7031 and 0.1083 in 2011 to 0.7545 and 0.1279 in 2018, respectively the weight of C₄ was only 0.028, and the multi-year average obstacle degree of C₁₂ was 0.1199, which was not enough to cause a significant negative impact on the WRCC; therefore, under the dual positive and negative impacts in 2019, the reason for the deterioration in the WRCC in Suzhou City was similar to that of Bozhou City—mainly due to the weakening in the support force.

(5)

The key obstacle factors affecting the enhancement in the WRCC in Bengbu City were C₁, C₂, C₄, C₅, and C₁₁ in 2011–2013 and 2015–2016, C₁, C₂, C₃, C₄, and C₁₁ in 2014, C₁, C₂, C₄, C₇, and C₁₁ in 2017–2018, and C₁, C₂, C₄, C₅, and C₈ in 2019. The multi-year average obstacle degree of C₅, C₇, C₈, and and C₁₁ from 2011–2019 were 0.7742, 0.6582, 0.4800, and 0.8695, respectively, and C₇ and C₈ can be considered not to be the key obstacle factors; therefore, the per capita water resources C₁, the annual water production modulus C₂, the vegetation coverage ratio C4, the utilization ratio of water resources C₅, and the population density C₁₁ were the key obstacle factors restricting the enhancement in the WRCC of Bengbu City, in which the support force, regulation force, and pressure force subsystems were four (3/4 × 100% = 75%), one (1/3 × 100% = 33.3%), and one (1/6 × 100% = 16.7%), respectively; meanwhile, C₁, as the first obstacle factor, appeared seven times (years) in nine years. Thus, the major reason for hindering the improvement in the WRCC of Huaibei City was an insufficient support force and a strong pressure force; meanwhile, the role of the regulation force was not obvious. The obstacles degree of the utilization ratio of water resources C₅, the per capita GDP C₆, the water consumption per CNY 10⁴ GDP C₉, the water consumption per 10⁴ yuan value-added by industry C₁₀, and the farmland irrigation quota C₁₃ have gradually decreased in Bengbu City from 2011 to 2018. In terms of the evaluation indexes, these indexes have improved to promote the carrying capacity. Therefore, rationally controlling the degree in water resource utilization, promoting economic growth through market regulation, improving the efficiency of industrial water resource utilization, and rationally controlling the agricultural irrigation quota to improve the efficiency in agricultural water resource utilization were powerful measures to improve the severe situation of the WRCC in Bengbu City. The major reason for the deterioration in the WRCC in Bengbu City in 2019 was the weakening of the support force and the strengthening of the pressure force. The per capita water resources C₁ and the annual water production modulus C₂ in the support subsystem deteriorated, and these obstacle degrees increased from 0.8659 and 0.7711 to 0.9532 and 0.9325 in 2018, respectively. The per capita daily domestic water consumption C₈ in the pressure force subsystem also deteriorated, and the obstacle degree increased from 0.4271 to 0.8757 in 2018.

(6)

The key obstacle factors affecting the enhancement in the WRCC in Fuyang City were C₁, C₂, C₃, C₄, and C₁₁ in 2011–2012, C₁, C₂, C₃, C₇, and C₁₁ in 2013, and C₁, C₂, C₃, C₄, and C₁₁ in 2014–2019. The utilization ratio of water resources C₅ has continued to evolve in a positive direction in 2011–2019, and decreased from 88.16% to 46.97%, and the corresponding obstacle degree decreased from 0.8661 to 0.3427, which was no longer the key obstacle factor. Therefore, the per capita water resources C₁, the annual water production modulus C₂, the per capita water supply C₃, the vegetation coverage ratio C₄, and the population density C₁₁ were the key obstacle factors limiting the improvement in the WRCC in Fuyang City. The obstacle degree of the utilization ratio of water resources C₅, the per capita GDP C₆, the water consumption ratio of the ecological environment C₇, the water consumption per CNY 10⁴ GDP C₉, the water consumption per 10⁴ yuan value-added by industry C₁₀, and the farmland irrigation quota C₁₃ in Fuyang City from 2011 to 2018 gradually decreased, i.e., the indexes improved, while the obstacle degrees of C₈ and C₁₂ gradually increased, i.e., the indexes worsened. In terms of the evaluation indexes, the water resources carrying state of Fuyang City developed to a better due to the overall decrease in the utilization ratio of water resources, the increase in the level of economic development, the increase in water consumption of the ecological environment year by year, and the decrease in the farmland irrigation quota, which was similar to that of Suzhou City. C₈ and C₁₂ gradually deteriorated due to the weights of C₈ and C₁₂ only being 0.0396 and 0.0632; meanwhile, the multi-year average obstacle degrees of C₈ and C₁₂ were 0.3830 and 0.1204, respectively, which did not play a dominant role in the direction of the evolution of the WRCC being insufficient; therefore, the utilization ratio of water resources C₅, the per capita GDP C₆, the water consumption ratio of the ecological environment C₇, the water consumption per CNY 10⁴ GDP C₉, the water consumption per 10⁴ yuan value-added by industry C₁₀, and the farmland irrigation quota C₁₃ drove the change in the WRCC in Fuyang City, which was the main driving factor for the development of the carrying capacity. In terms of the evaluation subsystem, the reason for the good development of the WRCC in Fuyang City was the reduction in the water resources carrying pressure force and the increment of the regulation force. The major reason for the deterioration of Fuyang City’s WRCC state in 2019 was the weakening of the support force and regulation force. The per capita water resources C₁ and the annual water production modulus C₂ in the support force subsystem deteriorated, and the obstacle degree increased from 0.9273 and 0.8827 in 2018 to 0.9769 and 0.9336, respectively, while the utilization ratio of water resources C₅ in the regulation force subsystem deteriorated, and the obstacle degree of it increased from 0.3427 in 2018 to 0.8715.

(7)

The key obstacle factors affecting the enhancement in the WRCC in Huainan City were C₁, C₂, C₄, C₅, and C₁₀ in 2011, C₁, C₂, C₄, C₅, and C₁₁ in 2012–2015 and 2019, and C₁, C₄, C₅, C₁₀, and C₁₁ in 2016–2018. The annual water production modulus C₂ continued to improve in 2011–2018, and the degree of obstacle decreased from 0.9075 to 0.1228, which can no longer constrain the carrying capacity leap, meaning that the key obstacle factors hindering the improvement in the WRCC in Huainan City were the per capita water resources C₁, the vegetation coverage ratio C₄, the utilization ratio of water resources C₅, the water consumption per 10⁴ yuan value-added by industry C₁₀, and the population density C₁₁, of which there were two indexes for the support force subsystem (2/4 × 100 = 50%), one index for the regulation force subsystem (1/3 × 100 = 33.3%), and two indexes for the pressure force subsystem (2/6 × 100 = 33.3%). In recent years, the situation of these three subsystems were more serious, which, together, caused the city’s water resources carrying capacity in the long term being in between the critically overloaded and overloaded states. Based on the year-by-year changes in the obstacle degree of the indexes to analyze the reasons for the changes in the carrying state, the obstacle degree of the per capita GDP C₆, the water consumption ratio of the ecological environment C₇, the water consumption per CNY 10⁴ GDP C₉, and the farmland irrigation quota C₁₃ in Huainan City from 2011–2018 gradually decreased, which meant that these indexes improved, while the obstacle degree of the population density C₁₁ gradually increased, which meant that the index worsened. It was found that on December 31, 2015, Shouxian County of Lu’an City was transferred to the jurisdiction of Huainan City, and its population density increased accordingly, and the diagnostic results of this city were in line with the reality. In terms of the evaluation indexes, the water resources carrying state of Huainan City has developed in a good direction due to the increase in the level of economic development, the increase in water consumption of the ecological environment year by year, a more coordinated relationship between economic development and water resource utilization, and the reduction in the farmland irrigation quota. The indexes of C₇ and C₁₃ have improved remarkably, decreasing to 0.5096 and 0.1403 from 0.8726 and 0.9002, respectively. Despite the fact that C₁₁ has been progressively deteriorating, the weight of C₁₁ is only 0.0792, which was not able to form a high pressure state, meaning that the WRCC of Huainan City was finally more inclined to the direction of improvement. In terms of the evaluation subsystem, the WRCC of Huainan City developed in the direction of improvement due to the pressure force decreasing and the regulation force increasing. The main reason for the deterioration in the WRCC of Huainan City in 2019 was the weakness of the support force and the regulation force and the increase in the pressure force. The per capita water resources C₁ and the annual water production modulus C₂ in the support force subsystem deteriorated, and these obstacle degrees increased to 0.9733 and 0.8944 from 0.8891 and 0.1228 in 2018, respectively. At the same time, the utilization ratio of water resources C₅ in the regulation force subsystem deteriorated, and the obstacle degree of it increased to 0.9726 from 0.8658 in 2018.

The existing obstacle degree model only considers the value of the index; its physical meaning is insufficient, the interpretability is not strong, its calculation process and results are static, and the diagnosis method is single [4,6,29,30]; meanwhile, the quantitative diagnosis method of the regional WRCC obstacle factor based on the connection number and TOPSIS constructed in this paper integrally takes into account the value of the WRCC evaluation index and the evaluation grade, and determines the ideal solution based on the set pair potential value of the evaluation index; the calculation is simple, the interpretability is strong, and the diagnosis results are reliable, which can quantitatively and dynamically identify the obstacle factors to the improvement in the WRCC and deeply analyze the reasons for the change in the carrying capacity, and the diagnosis results are consistent with the actual situation in Huaibei Plain, which provides a new perspective for dynamically determining the difference degree coefficient and accurately diagnosing the obstacle factors of the WRCC.

The key obstacles factors affecting the enhancement in the WRCC in the Huaibei Plain have certain differences and similarities in time and space, which were considered from two aspects: first, the difference and similarity of the obstacle factors in the same region under different years; second, the difference and similarity of the obstacle factors in different regions under the same years. Considering the length of this article, only Huaibei City in the years 2011, 2018, and 2019 were selected here to illustrate the difference and similarity of obstacle factors in the same region under different years; Huaibei City, Bozhou City, and Huainan City in the year 2018 were selected to illustrate the difference and similarity of obstacle factors in different regions under the same year, as shown in Figure 3 and Figure 4.

We observed the following from Figure 7 and Figure 8:

(1)

The key obstacle factors for Huaibei City in 2011 were the per capita water resources C₁, the annual water production modulus C₂, the per capita water supply C₃, the vegetation coverage ratio C₄, and the water consumption ratio of the ecological environment C₇; however, those for both 2018 and 2019 were the per capita water resources C₁, the annual water production modulus C₂, the per capita water supply C₃, the vegetation coverage ratio C₄, and the population density C₁₁, which showed that these obstacle factors continued to change over time, but in adjacent years tended to be similar.

(2)

In 2018, the key obstacle factors for Huaibei City and Bozhou City were the per capita water resources C₁, the annual water production modulus C₂, the per capita water supply C₃, the vegetation coverage ratio C₄, and the population density C₁₁. Huainan City had the per capita water resources C₁, he vegetation coverage ratio C₄, the utilization ratio of water resources C₅, the water consumption per 10⁴ yuan of value-added by industry C₁₀, and the population density C₁₁. Referring to the geographical location and administrative zoning map of Huaibei Plain, Anhui Province, Huaibei City was adjacent to Bozhou City and had similarities in terms of its water resources, hydrological element distribution, and other aspects; however, Huaibei City and Huainan City were significantly different in spatial location and presented spatial differences in the obstacle factors of the WRCC.

(3)

It can be seen that the method constructed in this paper not only provides a new intelligent method for the diagnosis of obstacle factors, but also identifies the distribution characteristics and spatial–temporal scale change rules of the key obstacle factors, which will help water resource managers to accurately regulate all elements of the water resource system, to realize the water resources carrying state to the better, and to promote the sustainable utilization of water resources.

5. Conclusions

This study focused on the quantitative and dynamic diagnosis of obstacle factors in the WRCC. Firstly, the mobility matrix was introduced to determine the CN, and the difference degree coefficient was calculated dynamically using the SSSPP and the triangular fuzzy number method, obtaining the connection number value, while at the same time using the level eigenvalue method and the attribute identification method to compute the WRCC grade. The obstacle degree of each index was quantitatively represented using the distance between the set pair potential value and the ideal solutions. The quantitative method of WRCC obstacle factor analysis based on the CN and TOPSIS was established, and finally, case studies were carried out for Huaibei Plain. The major conclusions were as follows:

(1)

The WRCC of the Huaibei Plain varied around grade 2 from 2011–2019, and its insufficient support force was the main reason why the WRCC of the Huaibei Plain has been in between the critically overloaded and overloaded states for a long period of time; the water resources carrying state improved from 2011–2018 due to the gradual improvement in the per capita GDP and the water consumption ratio of the ecological environment, etc., i.e., the regulation force was enhanced, and the WRCC of the Huaibei Plain was on a trend of deterioration in 2019, with certain factors, such as low precipitation in that year, reduced water resources, and a higher degree of water resource utilization, being listed as important reasons. Spatially, the WRCC of Fuyang City and Huainan City in 2011–2019 was worse than the other four cities, which was mainly related to the regional water resources, population size, GDP, and farmland irrigation quota. The above results are in good agreement with official data sources and the literature. In addition, the factors affecting the water resources carrying state of the Huaibei Plain changed in 2019, and the method constructed in this paper can dynamically identify the changes in the WRCC, which shows that it is effective to determine the connection number components based on the mobility matrix, as well as to dynamically determine the difference degree coefficient using the semi-partial subtraction set pair potential and triangular fuzzy number, and that this method is sensitive to the response under the changing conditions, which contributes to the precise management of water resources.

(2)

The key obstacles factors affecting the enhancement in the WRCC in the Huaibei Plain were the per capita water resources C₁, the annual water production modulus C₂, the per capita water supply C₃, the vegetation coverage ratio C₄, the utilization ratio of water resources C₅, the water consumption per 10⁴ yuan value-added by industry C₁₀, and the population density C₁₁. These indexes are in urgent need of manual regulation. The method in this paper overcame the static bias in the difference degree coefficient and the shortcomings of the previous diagnosis method, and the diagnosis results of the key obstacle factors are consistent with the water resources condition of Huaibei Plain, which provides a new perspective for the diagnosis of the WRCC obstacle factors.

(3)

In time and space, there were differences and similarities in the key obstacle factors. The obstacle factors continued to change over time, but in adjacent years tended to show similarities, such as Huaibei City in 2018 and 2019. In addition, the key obstacle factors in adjacent regions generally exhibited spatial similarity, such as Huaibei City and Bozhou City in 2018, while regions with significantly different spatial locations generally appeared as having a spatial difference, such as Huaibei City and Huainan City in 2018, which is consistent with the distribution of regional water resources and hydrological elements. It is thus clear that the method of this paper not only provides a new intelligent method for the diagnosis of obstacle factors, but also analyzes and identifies the distribution characteristics and spatial–temporal scale change rules of the obstacle factors.

(4)

The quantitative diagnosis method of obstacle factors of the WRCC based on the CN and TOPSIS adopted the idea of evaluation–diagnosis–regulation, which can quantitatively express the distance between each evaluation index and optimal solution and provide the obstacle degree, with a strong interpretation of physical meaning, a sensitive response under changing conditions, and evaluation and diagnosis results that were accurate and reliable. It not only affords a new insight into water resource management, but also offers an impactful way to accurately evaluate the resource and environmental carrying capacity, diagnose the obstacle factors, and enrich the analysis theory and the method of uncertainty systems that integrate comprehensive evaluation obstacle factor diagnosis regulation. For example, the method constructed in this paper can be applied to the diagnosis and analysis of the ecological carrying capacity, the agricultural green development level, and the regional resource and environmental carrying capacity obstacle factors. It is worth emphasizing that, when applying the method, attention should be paid to the validity of the selection of the evaluation indexes and the rationality of the evaluation results. The research method is only feasible when the research results are consistent with the actual situation.

(5)

The scale of our study is not fine enough due to the limitations of the raw data. In the future, we can also define the study area unit at a smaller scale, for example, at the county administrative level or in a kilometer grid (1 km × 1 km), and we can use spatial data analysis tools to evaluate the regional water resources carrying capacity and identify obstacle factors in a more refined way. We could also interface with the managers responsible for the water resources carrying capacity authorities regarding the evaluation results with a view to obtaining their approval.